Fog not darkening?

You're creating a function that you do not fire, which means that the fog will not be set. Also, the function is unnecessary as you could just set the FogColor in the code without the function.

Also, I'm not sure what the FogColor variable is so I'm going to assume that you've set the variable already. If you haven't set the FogColor variable, make sure you set that also, otherwise the FogColor will not change and the script will error.

Code:

local OtherworldPart = script.Parent
--No need for DarkenFog here.

function onTouch(part)
    local otherworldGui = Instance.new("ScreenGui")
        otherworldGui.Parent = game.Players.LocalPlayer.PlayerGui
    local otherworldText = Instance.new("TextLabel")
        otherworldText.Parent = otherworldGui
        otherworldText.Position = UDim2.new(0,0,0,0)
        otherworldText.Size = UDim2.new(0,200,0,50)
        otherworldText.BackgroundTransparency = 1
    game.Lighting.FogColor = FogColor.Value
        otherworldText.Text = "That's strange. It's getting darker."
        otherworldText.Font = Enum.Font.SourceSansBold
        otherworldText.FontSize = Enum.FontSize.Size36
        otherworldText.TextColor = BrickColor.new(255,255,255)
        otherworldText.TextStrokeTransparency = 0
    for i = 0, 1, .1 do
            otherworldText.TextTransparency = i
            wait(0.1)
    end
end

OtherworldPart.Touched:connect(onTouch)

I hope my answer helped you. If it did, be sure to accept it.